1 Introduction

Isothermal spheres play an important role in astrophysics. They were initially introduced in the context of stellar structure (Chandrasekhar 1942) when composite configurations of stars consisting of an isothermal core and a polytropic envelope were constructed and studied. On larger scales, they were applied to stellar systems such as globular clusters and elliptical galaxies (Binney & Tremaine 1987). The age of globular clusters is such that their isothermal structure is due to a succession of encounters between stars that lead to an equipartition of energy, like in an ordinary gas. This statistical mechanics prediction has been confirmed by direct observations of globular clusters and is well reproduced by King's models that incorporate a truncation in the distribution function so as to account for tidal effects. In the case of elliptical galaxies (and possibly other collisionless self-gravitating systems like massive neutrinos in Dark Matter models), the isothermal distribution is a result of a "violent relaxation'' by phase mixing (Lynden-Bell 1967). In that case, there is no segregation by mass contrary to the collisional relaxation. In general, violent relaxation is incomplete and the distribution function must be modified at high energies. However, the distribution function that prevails in the inner regions of elliptical galaxies is isothermal and this is an important ingredient to understand de Vaucouleurs' R^1/4 law (Hjorth & Madsen 1993). Isothermal distributions are also appropriate to describe the cold interstellar medium where the temperature is imposed by the cosmic background radiation at $T\sim 3$ K in the outer parts of galaxies, devoid of any star and heating sources (Pfenniger & Combes 1994; de Vega et al. 1996). They can also be of interest in cosmology to understand the fractal structure of the universe (Saslaw & Hamilton 1984; de Vega et al. 1998).

The stability of isothermal gaseous spheres in Newtonian gravity has been investigated by Antonov (1962), Lynden-Bell & Wood (1968) and Katz (1978) by using thermodynamical arguments and topological properties of the equilibrium phase diagram by analogy with Poincaré's theory of linear series of equilibrium. They showed in particular that, if the energy or the temperature are below a certain threshold, no hydrostatic equilibrium can exist. In that case, the system is expected to undergo a phase transition and collapse. This is called "gravothermal catastrophe'' when the energy is kept fixed (microcanonical ensemble) and "isothermal collapse'' when the system evolves at a fixed temperature (canonical ensemble). The thermodynamics of self-gravitating systems was reconsidered by Padmanabhan (1989) who calculated explicitly the second order variations of entropy and reduced the problem of stability to an eigenvalue equation. His method was applied in the canonical ensemble by Chavanis (2001) who showed in addition the equivalence between thermodynamical stability and dynamical stability based on the Navier-Stokes equations (Jeans problem). Similar studies have been performed by Semelin et al. (1999, 2000) and de Vega & Sanchez (2001) by using field theoretical methods. As shown by Padmanabhan (1989) and Chavanis (2001), the problem of stability can be studied without approximation almost analytically (or with graphical constructions) by making use explicitly of the Milne variables introduced long ago in the context of stellar structure (Chandrasekhar 1942).

We show in the present paper that these methods naturally extend to the context of general relativity. Isothermal gaseous spheres (in the sense given below) have been only poorly studied in general relativity despite their similarity with classical isothermal spheres. The most extended study that we have found is the contribution of Chandrasekhar at the conference given in the honour of J. L. Synge in 1972. His work was completed by Yabushita (1973, 1974) who considered the stability of a relativistic isothermal gas surrounded by an external envelope imposing a constant pressure (an extension of the classical Bonnor 1956 problem). In order to make the link with the works of Antonov (1962) and Lynden-Bell & Wood (1968) in Newtonian gravity, we shall consider the situation in which the volume is fixed instead of the pressure. This study can have direct applications to the stability of neutron stars since the equation of state that prevails in the central region of these highly relativistic objects is very close to the "isothermal'' one.

In Sect. 2, we consider the stability of isothermal gas spheres in the framework of Newtonian gravity but taking into account special relativity. This is a first attempt to introduce relativistic effects in the classical Antonov problem. In fact, as is well-known, special relativity does not modify the equation of state for a perfect gas (Chandrasekhar 1942). Only does it alter the onset of instability. We find that the critical energy below which no hydrostatic equilibrium is possible depends on a relativistic parameter $\mu$ defined as the ratio between the size of the domain R and a "classical'' Schwarzschild radius $R_{\rm S}^{\rm class}=2GM/c^{2}$, where M is the total mass of the system. In the classical limit $R\gg R_{\rm S}^{\rm class}$, we recover the Antonov result $E_{\rm c}=-0.335GM^{2}/R$ but when relativistic corrections are included we find that the critical energy is increased, i.e., instability occurs sooner than in the classical case. The density perturbation profile that triggers the instability is calculated explicitly. For $\mu\rightarrow +\infty$, it presents a "core-halo'' structure but relativistic effects tend to reduce the extent of the halo. When the system is maintained at a fixed temperature instead of a fixed energy (canonical description), the results are unchanged with respect to the Newtonian case.

Of course, when the Schwarzschild radius becomes comparable to the size of the system, general relativistic effects must be taken into account. This problem is treated in detail in Sect. 3. We consider a simple equation of state $p=q\epsilon$(where q is a constant) which generalizes the equation of state for isothermal spheres in the Newtonian context (the classical limit is recovered for $q\rightarrow 0$). Since the equations governing equilibrium have the same structure and the same properties as in the classical case, we shall say that the system is "isothermal'' (following the terminology of Chandrasekhar 1972), although this equation of state does not correspond to thermal equilibrium in a strict sense (see Sect. 3.2). True statistical equilibria in general relativity have been investigated by Katz et al. (1975) and they are characterized by a non uniform temperature (because of the gravitational redshift). However, systems described by the equation of state $p=q\epsilon$ are numerous in nature and they include for example the important case of neutron cores which are usually modeled as an assembly of cold degenerate fermions for which q=1/3. This simple isothermal equation of state is the high-density limit of more general equations of state usually considered for neutron stars (Oppenheimer & Volkoff 1939; Meltzer & Thorne 1966). Quite remarkably, our simple "box'' model is able to reproduce the main properties of these objects. This suggests that the structure of neutron stars is due intrinsically to their isothermal cores and not to the details of their envelopes (which are not well-known). In Sect. 3.5, we show that the mass-density diagram displays an infinity of mass peaks and that isothermal spheres exist only below a limiting mass corresponding to the first peak. In Sect. 3.6, we show that the mass-radius diagram has a spiral behavior similar to the one observed for neutron stars. Using an equation of pulsation derived by Yabushita (1973), we demonstrate analytically that the series of equilibrium becomes unstable precisely at the point of maximum mass (like in the study of Misner & Zapolsky 1964) and that new modes of instability correspond to secondary mass peaks. We obtain the same stability criterion from energy considerations based on the binding energy E=M-Nmc². Said differently, these results indicate that, for a fixed mass, the system becomes unstable when its radius is smaller than a multiple of the Schwarzschild radius, a property consistent with Chandrasekhar's (1964) general theory. The perturbation profiles of density and velocity that trigger the instability are calculated explicitly and expressed in terms of the Milne variables. They do not present a "core-halo'' structure. All these properties are strikingly similar to those obtained for classical isothermal spheres in the canonical ensemble (Chavanis 2001). This completes the analogy between isothermal spheres and neutron stars investigated by Yabushita (1974).